Nuclearity of semigroup C*-algebras and the connection to amenability

Abstract

We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C*-algebras.

arXiv comment42 pages; revised version, corrected typos

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Cited by in corpus (8)

• On C*-algebras associated to right LCM semigroups
• Nica-Toeplitz algebras associated with product systems over right LCM semigroups
• K-theory for semigroup C*-algebras and partial crossed products
• On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids
• Co-universality and controlled maps on product systems over right LCM-semigroups
• On the K-theory of crossed products by automorphic semigroup actions
• On K-theoretic invariants of semigroup C*-algebras attached to number fields
• A new approach to recent constructions of C*-algebras from modular index theory